## On the prime radical of a module over a commutative ring.(English)Zbl 0776.13003

All rings will be commutative with identity and all modules will be unitary. The prime radical of a module $$M$$ over a ring $$R$$ is the intersection of $$M$$ with all prime submodules of $$M$$. We shall denote by $$W(M)$$ the set of elements of $$M$$ which may be written in the form $$r_ 1m_ 1+\cdots+r_ nm_ n$$, where $$n$$ is a positive integer, $$r_ i \in R$$, $$m_ i \in M$$, and $$r^ k_ im_ i=0\;(1 \leq i \leq N)$$, for some positive integer $$k$$. Among other results the authors prove the following lemma:
Let $$R$$ be a Noetherian domain of dimension 1 and let $$M$$ be any $$R$$- module. Then $$\text{rad} M=\bigcup \text{rad} L$$, where the union is taken over all finitely generated submodules $$L$$ of $$M$$.
Corollary. Let $$R$$ be any ring and $$M$$ any projective $$R$$-module. Then $$\text{rad} M=W(M)$$.
Theorem. Let $$R$$ be a Dedekind domain and $$M$$ any $$R$$-module. Then $$\text{rad} M=W(M)$$.
An $$R$$-module $$M$$ “satisfies the radical formula” if $$\text{rad} (M/N)=W(M/N)$$ for any submodule $$N$$ of $$M$$. If every $$R$$-module satisfies the radical formula we say that $$R$$ does (STRF). The authors prove that for any $$R$$-module $$M$$, $$M$$ STRF is equivalent to every semiprime submodule of $$M$$ being an intersection of prime submodules of $$M$$ and $$W[(M/W(M)]=0$$. – An example of a semiprime submodule which is not the intersection of prime submodules is given to answer a query of Dauns, and several equivalences to STRF are given for a Noetherian UFD.

### MSC:

 13A10 Radical theory on commutative rings (MSC2000) 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations